Download Physics for Scientists & Engineers 2

Review
! The index of refraction of an optical material, n, is given by
n=
Physics for Scientists &
Engineers 2
c
v
• c is the speed of light in a vacuum
• v is the speed of light in the optical material
• n ! 1, nvacuum =1
Spring Semester 2005
• We will use nair = 1
Lecture 38
! The Law of Refraction or Snell’s Law
can be expressed as
n1 sin !1 = n2 sin ! 2
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Lenses
! The angle at which the light rays cross the boundary is different
along the boundary, so the refracted angle is different at
different points along the boundary
! A curved boundary between two optically transparent media is
called a lens
! If the front surface of the lens is part of the surface of a sphere with
radius R1 and the back surface of the lens is part of the surface of a
sphere with radius R2, then we can calculate the focal length f of the
lens using the lens-makers formula
" 1
1
1%
= ( n ! 1) $ ! '
f
# R1 R2 &
! Note that in this equation R2 is negative because it has the opposite
curvature from the front surface
! Light rays that are initially parallel before they strike the
boundary are refracted in different directions depending on the
part of the lens they strike
! If we have a lens with the same radii on the front and back of the lens
so that R1 = R2 = R, we get
1 2(n ! 1)
=
f
R
! Depending of the shape of the lens, the light rays can be focused
or caused to diverge
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Lens Maker Formula
! When light is refracted crossing a curved boundary between two
different media, the light rays follow the law of refraction at
each point on the boundary
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Lens Focal Lengths
Convex Lenses
! A convex lens is shaped such that parallel rays will be focused by
refraction at the focal distance f from the center of the lens
! Unlike mirrors, lenses have a focal length on both sides
! Light can pass through lenses while
mirrors reflect the light allowing
no light on the opposite side
! In the drawing on the right, a light ray is incident
on a convex glass lens
! At the surface of the lens, the light ray is
refracted toward the normal
Convex (converging) lens
! When the ray leaves the lens, it is
refracted away from the normal
! The focal length of a
convex (converging) lens is defined
to be positive
! The focal length of a
concave (diverging) lens is
defined to be negative.
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! Let us now study the case of several
horizontal light rays incident on a
convex lens
! These rays will be focused to a point a
distance f from the center of the lens on
the opposite side from the incident rays
Concave (diverging) lens
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Thin Lenses
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Images with Convex Lenses
! The rays enter the lens, get refracted at the surface, traverse the
lens in a straight line, and get refracted when they exit the lens
! Convex lenses can be used to form images
! We show the geometric construction of the formation of an image using
a convex lens with focal length f
! In the third frame, we represent the thin lens approximation by
drawing a black dotted line at the center of the lens
! Instead of following the detailed trajectory of the light rays inside the
lens, we draw the incident rays to the centerline and then on to the
focal point
! This real-life lens is a thick lens
and there is displacement
between the refraction at the
entrance and exit surfaces
! We place an object standing on the optical axis represented by the
green arrow
! We will treat all lenses as thin
lenses and treat the lens as a line
at which refraction takes place
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! This object has a height ho and is located a distance do from the center
of the lens such that do > f
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Images with Convex Lenses (2)
Images with Convex Lenses (3)
! We start with a ray along the optical axis of the lens that passes
straight through the lens that defines the bottom of the image.
! A second ray is then drawn from the top of the object parallel to
the optical axis
• This ray is focused through the focal point on the other side of the lens
! The second ray is drawn from the top of the object parallel to the
optical axis and is focused through the focal point on the opposite side
! A fourth line is drawn such that
it originated from the focal point
on the same side of the lens and
is then focused parallel to the
optical axis
! A fourth ray is drawn from
the top of the object
through the focal point on
the same side of the lens
that is then directed
parallel to the optical axis
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! The first ray again is drawn from the bottom of the object along the
optical axis
! A third ray is drawn through the
center of the lens
! A third ray is drawn through the center of the lens that is not
refracted in the thin lens approximation
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! Now let us consider the image formed by an object with height placed
a distance from the center of the lens such that do < f
• These three rays are diverging
! A virtual image is located on the
same side of the lens as the object
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Concave Lenses
! Assume that a light ray parallel to
the optical axis is incident on a
concave glass lens
! At the surface of the lens, the
light rays are refracted toward
the normal
! When the rays leave the lens, they are refracted away from the normal
as shown
! The extrapolated line shown as a red and black dashed line that points
to the focal point on the same side of the lens as the incident ray
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Concave Lenses (2)
! A concave lens is shaped such that parallel rays will be caused to
diverge by refraction such that their extrapolation would intersect at a
focal distance from the center of the lens on the same side of the lens
as the rays are incident
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!
Let us now study several horizontal light rays incident on a concave lens
!
After passing through the lens, the rays will diverge such that their
extrapolations intersect at a point a distance f from the center of the lens on
the same side of the lens as the incident rays
!
To the right is a concave lens with five
parallel lines of light incident in the surface
of a concave lens from the left
!
In the second panel we have drawn red lines
representing light rays
!
We can see that the light rays diverge after
passing through the lens
!
We have drawn red and black dashed lines to show the extrapolation of the
diverging rays
!
The extrapolated rays intersect a focal length away from the center of the lens
!
In the third panel we draw the diverging rays using the thin lens approximation
where the incident rays are drawn to the center of the lens
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Images formed with Concave Lenses
The Lens equation
! The images formed by lenses are described by the lens equation
!
Here we show the formation of an image using a concave lens
!
We place an object standing on the optical axis represented by the green arrow
!
This object has a height ho and is located a distance do from the center of the
lens such that do > f
!
!
We again start with a ray along the
optical axis of the lens that passes
straight through the lens that defines
the bottom of the image
• We define the focal length f of a convex lens to be positive and the focal
length of a concave lens to be negative
This ray is refracted such that its
extrapolation of the diverging ray passes
through the focal point on the other side of the lens
!
A third ray is drawn through the center of the lens that
is not refracted in the thin lens approximation
!
The image formed is virtual, upright, and reduced
•
• We define the object distance do to be positive
• If the image is on the opposite side of the lens from the object, the image
distance di is positive and the image is real
• If the image is on the same side of the lens as the object, the image
distance di is negative and the image is virtual
This ray is extrapolated back along its original path
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! T
his equation is the same relationship between focal length, image
distance, and object distance that we had found for mirrors
! To treat all possible cases for lenses, we must define some conventions
for distances and heights
A second ray is then drawn from the top
of the object parallel to the optical axis
•
1 1 1
+ =
d o di f
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• If the image is upright, then hi is positive and if the image is inverted, hi is
negative
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Special Cases
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Magnification for Lenses
! For a convex lens, we find that for do > f we always get a real, inverted
image formed on the opposite side of the lens
! For a convex lens and do < f , we always get a virtual, upright, and
enlarged image on the same side of the lens as the object
! The special cases for do > f for a convex lens are
! The magnification m of a lens is defined the same as as for
a mirror
d
h
m=! i =! i
do
ho
• do is the object distance
• ho is the object height
Case
f < do < 2 f
do = 2 f
do > 2 f
Type
Real
Real
Real
Direction
Inverted
Inverted
Inverted
• di is the image distance
Magnification
Enlarged
Same size
Reduced
• hi is the image height
! If |m| > 1, the image is enlarged
! If |m| < 1, the image is reduced
! If m < 0, the image is inverted
! For concave lenses, we always get an image that is virtual, upright, and
reduced in size
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! If m > 0, the image is upright
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Power of Lenses
Example: Image formed by Convex Lens
! The focal length of a converging lens is
21 cm. An object is located a distance
do = 32 cm from the center of the lens.
! The power of a lens is often quoted rather than its folca
length
! The power of a lens, D (diopters), is given by the equation
! Where is the image?
1m
D=
f
! What is the magnification of the image?
convex lens ! f is positive
1 1 1
d f
+ =
! di = o
d o di f
do " f
! For example, common reading glasses have a power of
D = 1.5 diopters
di =
! The focal length of these glasses is
f=
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image will be located 61 cm to the right of the lens
d
h
0.61 m
m=" i =" i ="
= "2.9 ! enlarged, inverted
do
ho
0.21 m
1m
= 0.67 m
1.5 diopters
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( 0.320 m ) ( 0.210 m ) = 0.61 m
( 0.320 m ) " ( 0.210 m )
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